Mudstone, serving as a gas-sealing medium in coal seams, exhibits low permeability that can lead to gas accumulation and the formation of high gas-risk zones. The competitive synergy among the principal stresses significantly impacts mudstone permeability, thereby complicating accurate permeability assessments in engineering applications. In this study, true triaxial fluid–solid coupling experiments were conducted under staged multiaxial loading and single principal stress control to elucidate the permeability evolution of mudstone. The results indicate that permeability is most sensitive to σ3, whereas σ2 modulates permeability primarily by regulating the loading rate and suppressing amplitude. Based on the elastic strain energy analysis, it is found that the uniform compaction of σ1 has a more limited effect on the alteration of mudstone microfracture structure and permeability, while σ3 triggers a cascading effect of fracture opening and closing due to its direct regulation of fracture opening and closing. A slight variation in σ3 triggers a pronounced fluctuation in permeability, which is caused by the conversion of the reduced strain energy into tension fractures in that direction. Further single-principal-stress experiments confirm the dynamic competition in energy distribution between σ1 and σ3. On this basis, the study introduces an innovative anisotropic permeability model that integrates a sensitivity coefficient and an energy competition index, offering a theoretical framework for regulating gas flow in underground coal seams.